… extends to general étale stacks, namely that any étale stack X … effective étale stack Y arises
uniquely from some étale stack Z … of an étale stack Z out of an effective étale stack Y equipped …

… étale differentiable stacks and orbifolds on the one hand, and higher algebraic Deligne-Mumford
stacks … Deligne-Mumford iff it is in the essential image of the étale prolongation functor …

… When restricting to the case of étale stacks, we are aided by our … An ∞-stack X on Mfd is
an étale differentiable ∞-stack if and only if it is in the essential image of the étale prolongation …

… notion of a stack over another stack and the distinction between large and small stacks. We
… Following Moerdijk [Moe02], we define an orbifold as a smooth proper étale groupoid. As in …

… Etale differentiable stacks are differentiable stacks which are mild … higher étale differentiable
stacks. Morally, such a higher … called the étale prolongation functor. This functor gives a …

… etale van Kampen theorems are proven by using known, results about the homotopy theory
of stacks and the characterization of etale … site C then the prolongation functor i* from the …

… morally, this is by Tate’s theorem on the uniqueness of prolongations of p-divisible groups). …
While we do not want to develop a general theory of the étale π0 groups of algebraic stacks, …

… smooth base stack, we introduce a formalism of stacks of relative … Thus, composing the PDE
with the jet prolongation … classes of generalized étale and submersive maps of stacks and …

… moduli stack C of rank two projective Breuil–Kisin modules, and the moduli stack R of étale
… , this is by Tate’s theorem on the uniqueness of prolongations of p-divisible groups). These …

… calculation of the shape of the Haefliger stack seems almost inevitable using the calculus of
… For any r-times differentiable étale stack X its image under j! is usually likewise denoted by X…